Machine Design Databook Episode 3 part 13 pot
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The expression for deflection surface of plate which
satisfies the boundary conditions Eq. (27-389) and
Eq. (27-388b)
Substituting Eq. (27-390) in Eq. (27-388b) and solving
for C
Equation (27-390) for w becomes
The expression for M
x
, M
y
and M
xy
The maximum deflection and bending moments,
which occur at midpoint of plate
The maximum deflection and bending moments for a
square plate
The shearing forces from Eqs. (27-368)
w ¼ C sin
x
a
sin
y
b
ð27-390Þ
C ¼
q
0
4
D
1
a
2
þ
1
b
2
2
ð27-390aÞ
w ¼
q
0
4
D
1
a
2
þ
1
b
2
sin
x
a
sin
y
b
ð27-390bÞ
M
x
¼
q
0
2
1
a
2
þ
1
b
2
2
1
a
2
þ
v
b
2
sin
x
a
sin
y
b
ð27-391aÞ
M
y
¼
q
0
2
1
a
2
þ
1
b
2
2
v
a
2
þ
1
b
2
sin
x
a
sin
y
b
ð27-391bÞ
M
xy
¼
q
0
ð1 À vÞ
2
1
a
2
þ
1
b
2
2
ab
sin
x
a
sin
y
b
ð27-391cÞ
w
max
¼
q
0
4
D
1
a
2
þ
1
b
2
2
ð27-392aÞ
M
x max
¼
q
0
4
D
1
a
2
þ
1
b
2
2
1
a
2
þ
v
b
2
ð27-392bÞ
M
y max
¼
q
0
2
1
a
2
þ
1
b
2
2
v
a
2
þ
1
b
2
ð27-392cÞ
w
max
¼
q
0
a
4
4
4
D
; M
x max
¼ M
y max
¼
ð1 þ vÞq
0
a
2
4
2
ð27-393Þ
Q
x
¼
q
0
a
1
a
2
þ
1
b
2
cos
x
a
sin
y
b
ð27-394aÞ
Q
y
¼
q
0
b
1
a
2
þ
1
b
2
sin
x
a
cos
y
b
ð27-394bÞ
Particular Formula
27.82 CHAPTER TWENTY-SEVEN
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APPLIED ELASTICITY
The reactive forces at the support edges at x ¼ a and
y ¼ b respectively
The resultant reaction concentrated at the corners of
the plate
The total pressure on all four edges of plate
The four corners reactions, which are equal due to
symmetry
The maximum bending stress if a > b is due to M
y
which is greater than M
x
Using Eq. (27-395d), the expression for maximum
shear stress which is at the middle of the longer side
of the plate
V
x
¼
Q
x
À
@M
xy
@y
x ¼a
ð27-395aÞ
V
x
¼À
q
0
a
1
a
2
þ
1
b
2
2
1
a
2
þ
2 À v
b
2
sin
y
b
ð27-395bÞ
V
y
¼
Q
y
À
@M
xy
@x
y ¼b
ð27-395cÞ
V
y
¼À
q
0
b
1
a
2
þ
1
b
2
2
1
b
2
þ
2 À v
a
2
sin
x
a
ð27-395dÞ
R ¼ 2ðM
xy
Þ
x ¼a
y ¼b
¼
2q
0
ð1 À vÞ
2
ab
1
a
2
þ
1
b
2
2
ð27-396Þ
2
ð
b
0
v
x
dy þ 2
ð
a
0
v
y
dx
¼
4q
0
ab
4
þ
8q
0
ð1 À vÞ
2
ab
1
a
2
þ
1
b
2
2
ð27-397Þ
ÁR ¼
8q
0
ð1 À vÞ
2
ab
1
a
2
þ
1
b
2
2
which is the second term on the right hand side of Eq.
(27-397)
y max
¼
6M
y max
h
2
¼
6q
0
2
h
2
1
a
2
þ
1
b
2
2
v
a
2
þ
1
b
2
ð27-398Þ
ð
yz
Þ
max
¼
3q
0
2bh
1
a
2
þ
1
b
2
2
1
b
2
þ
2 À v
a
2
ð27-399Þ
Particular Formula
APPLIED ELASTICITY
27.83
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.
APPLIED ELASTICITY
